Cardinal Invariants and the Collapse of the Continuum by Sacks Forcing

Journal of Symbolic Logic 73 (2):711 - 727 (2008)
Abstract
We study cardinal invariants of systems of meager hereditary families of subsets of ω connected with the collapse of the continuum by Sacks forcing S and we obtain a cardinal invariant yω such that S collapses the continuum to yω and y ≤ yω ≤ b. Applying the Baumgartner-Dordal theorem on preservation of eventually narrow sequences we obtain the consistency of y = yω < b. We define two relations $\leq _{0}^{\ast}$ and $\leq _{1}^{\ast}$ on the set $(^{\omega}\omega)_{{\rm Fin}}$ of finite-to-one functions which are Tukey equivalent to the eventual dominance relation of functions such that if $\germ{F}\subseteq (^{\omega}\omega)_{Fin}$ is $\leq _{1}^{\ast}$ -unbounded, well-ordered by $\leq _{1}^{\ast}$ , and not $\leq _{0}^{\ast}$ -dominating, then there is a nonmeager p-ideal. The existence of such a system F follows from Martin's axiom. This is an analogue of the results of [3], [9, 10] for increasing functions
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DOI 10.2178/jsl/1208359068
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References found in this work BETA
Iterated Perfect-Set Forcing.James E. Baumgartner & Richard Laver - 1979 - Annals of Mathematical Logic 17 (3):271-288.
Sacks Forcing, Laver Forcing, and Martin's Axiom.Haim Judah, Arnold W. Miller & Saharon Shelah - 1992 - Archive for Mathematical Logic 31 (3):145-161.
More Forcing Notions Imply Diamond.Andrzej Rosłanowski & Saharon Shelah - 1996 - Archive for Mathematical Logic 35 (5-6):299-313.

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